The word matrix will, in this paper, always connote a matrix with non-negative elements, having in general m rows and n columns. In conformity with the definition in (4), two matrices will be said to have the same pattern if the entry in any row or column is zero or not according as the corresponding entry of the other is zero or not. The symbols ρi and σj will stand for the respective phrases “ith row-sum“ and “jth column-sum” of the matrix under consideration. r1 … , rm; c1, … , cn is a set of positive numbers. It will be said to be consistent for the pattern of an m × n matrix, if there exists a matrix of that pattern for which ρi = rj and σj = cj for all i and j.

The notions of monotone bases and bases of subspaces are well known in a normed linear space setting and have obvious extensions to pseudo-metrizable linear topological spaces. In this paper, these notions are extended to arbitrary linear topological spaces. The principal result gives a list of properties that are equivalent to a sequence (Mi) of complete subspaces being an e-Schauder basis of subspaces for the closed linear span of . A corollary of this theorem is the fact that an e-Schauder basis for a dense subspace of a linear topological space is an e-Schauder basis for the whole space.

The usual definition of a finite probabilistic automaton (4; 5) does not involve any outputs. In this paper, such automata are referred to as Rabin automata. Various other types of probabilistic automata are obtained by introducing output functions. Similarly, as in connection with finite deterministic automata, the distinction between Moore automata and Mealy automata can then be made. Rabin automata may be regarded as automata where the output function is deterministic. One can also consider the case where the transition function is deterministic.

One of the really significant advances in stable homotopy theory has been the Adams spectral sequence (see (1) for a general discussion). To date there has been no useful general way to obtain differentials in this spectral sequence. There is a general feeling that these differentials come about because of some geometric fact which forces a difference between the ring structure imposed on π*(S0), the stable homotopy ring, by the spectral sequence and its own natural one.

Recently several papers (11; 12; 13; 14) have been published in which it is shown that a Moore space (normal, in one case) is metrizable if it has the peripheral version (in the sense defined below) of a certain covering property that was known to imply metrizability of Moore spaces. Each of these metrization theorems can be proved more easily by using a slight variation of the appropriate standard proof to show that such a space is collectionwise normal and hence (2, Theorem 10) metrizable. But this approach, as well as that followed in (11 ; 12; 13 ; 14), obscures the point that, in Moore spaces and in more general settings, the peripheral versions of these covering properties imply the covering properties.

Let Q be a quasi-ordered set with respect to ⩽ ; that is, the order ⩽ is reflexive and transitive. An element a of Q is called maximal (minimal) if

a is called greatest (smallest) if

Obviously a greatest (smallest) element is maximal (minimal). A greatest (smallest) element in a partially ordered set is unique, but it is not necessarily unique in a quasi-ordered set.

In previous papers (7; 8), we have investigated certain properties of general function algebras which may be regarded as generalizations or analogues of familiar results in the theory of analytic functions of several complex variables. This investigation is continued and expanded in the present paper. The main results concern a notion of holomorphic convexity for the general situation. We also extend somewhat several of the results obtained in the earlier papers.

Suppose that G is a locally compact abelian group, u an element of infinite order, and w a complex number of modulus 1. It is a familiar fact that there is a complex homomorphism Ψ of the measure algebra M of G, which maps ϵu (the unit mass concentrated at u) to w. Beyond this, one may specify an element μ of M, and require a homomorphism Ψ which does not annihilate μ. The resolution of this problem leads to an abstract lemma on measurable transformations, derived in some generality in the first section.

Let P be a finite set with p objects oj, j = 1, 2, … , p, and let {Si}, i = 1, 2, … , n, be a family of n subsets of P. The incidence matrix A = (aij) for the family {Si} is defined by the rules: aij = 1 if 0j, ∈ Si and aij = 0 if 0j ∉ Si. Then, if AAT = B = (bij) (where AT denotes the transpose of A), it is easy to see that bij = |Si ⋂ Sj|, i = 1, … , n, j = 1, … , n, so that the elements of B are integers with bii ⩾ bij ⩾ 0.

Marston Morse and William Transue (4; 5; 6) have developed a theory of seminormed Banach spaces of complex or real-valued functions called MT spaces. For E a locally compact space, the space of real-valued functions on E, and K the space of real-valued functions on E with compact supports, a real MT space A is a vector subspace of containing K and provided with a suitable seminorm NA.

Let L be any simple finite-dimensional Lie algebra (defined over the field K of complex numbers). Cartan's theory of weights is used to define sets of (algebraic) representations of L that can be characterized in terms of left ideals of the universal enveloping algebra of L. These representations, called standard, generalize irreducible representations that possess a dominant weight. The newly obtained representations are all infinite-dimensional. Their study is initiated here by obtaining a partial solution to the problem of characterizing them by means of sequences of elements in K.

In 1945 Ingham (3) proved the following Tauberian theorem: if ƒ is a non-decreasing, non-negative function on [1, ∞) and

1

then ƒ(x) ∼ cx. His proof is based on the non-vanishing of the Riemann zeta-function, ζ (s), on the line , and uses Pitt's form of Wiener's Tauberian theorem; (see, e.g., 5, Theorem 109, p. 211).

In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition

1.1

and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.

A mapping ϕ from one group, G, into another, H, is said to be a semi-homomorphism of G if ϕ(aba) = ϕ(a) ϕ(b) ϕa) for all a, b ∊G. Clearly any homomorphism or anti-homomorphism is a semi-homomorphism; the converse, however, need not be true in general. It is perfectly clear what one intends by a semi-isomorphism or semi-automorphism.

Our purpose here is to show that for a rather general situation a semi-homomorphism turns out to be a homomorphism or an anti-homomorphism. In (2) we proved that any semi-automorphism of a simple group which contains an element of order 4 must automatically be either an automorphism or an anti-automorphism.

Let be a completely regular Hausdorff space and C() the algebra of continuous real-valued functions on . In attempts to characterize abstractly those algebras that are isomorphic to C() for some , one produces subalgebras of C() which: (a) contain the constant functions, (b) separate points and closed sets in , (c) are closed under uniform convergence, and (d) are closed under inversion in C() (see, for example, (2; 5)).

Non-linear problems have been studied by Krasnoselski, Browder, and others; in fact Browder and independently Kirk (cf., 1; 5) have proved the following remarkable theorem: let X be a uniformly convex Banach space, U a non-expansive mapping of a bounded closed convex subset C of X into C, i.e., ||Ux — Uy|| ⩽ ||x — y|| for x, y ∊ C; then U has a fixed point in C. The aim of this paper is to give some existence theorems for non-linear functional equations in uniformly convex Banach spaces. Similar results may be found in (3 ; 6).

Nakayama proposed to classify finite-dimensional algebras R over a field according to how long an exact sequence

of projective and injective R-R-bimodules Xi they allow. He conjectured that if there exists an infinite sequence of this type, then R must be quasi-Frobenius; and he proved this when R is generalized uniserial (17).

This paper deals with several related properties of bounded summability fields of regular, real matrices. For a matrix A = (ank) and a sequence x = {xn}, we write formally

We denote by m the space of bounded real sequences, and by A* the bounded summability field

of A. The strong summability field of A is the set

If X is a space and , the kth deleted product space Xk* of X is the topological product X ⨯ X ⨯ . . . ⨯ X of copies of X minus the set of all points of the form (x, x, . . . , x), where x ∊ X. In (4), the author shows that the homology groups of Xk*, where X is a tree, produce as much information about trees as counting the orders of vertices.

The object of this note is to report on some of the properties of a class of operators induced by inner functions. If m is normalized Lebesgue measure on the unit circle X in the complex plane and Cϕ is an inner function (a complex function on X of unit modulus almost everywhere whose Poisson integral is a non-constant holomorphic function in the open unit disk), then an operator Cϕ on L2(m) is defined by